Introduction to Probability and Statistics for Engineers and Scientists

568 Chapter 13:Quality Control

To determine the variance, we again use Equation 13.6.2:

Var(Wt)=

σ^2

n

{

α^2 +[α(1−α)]^2 +[α(1−α)^2 ]^2 +···+[α(1−α)t−^1 ]^2

}

=

σ^2

n

α^2 [ 1 +β+β^2 +···+βt−^1 ] whereβ=(1−α)^2

=

σ^2 α^2 [ 1 −(1−α)^2 t]

n[ 1 −(1−α)^2 ]

=

σ^2 α[ 1 −(1−α)^2 t]

n(2−α)

Hence, whentis large we see that, provided that the process has remained in control
throughout,

E[Wt]=μ

Var(Wt)≈

σ^2 α

n(2−α)

since (1−α)^2 t≈ 0

Thus, the upper and lower control limits forWtare given by

UCL=μ+ 3 σ

√

α

n(2−α)

LCL=μ− 3 σ

√

α

n(2−α)

Note that the preceding control limits are the same as those in a moving-average control
chart with spank(after the initialkvalues) when

3 σ

√

nk

= 3 σ

√

α

n(2−α)

or, equivalently, when

k=

2 −α

α

or α=

2

k+ 1

EXAMPLE 13.6b A repair shop will send a worker to a caller’s home to repair electronic
equipment. Upon receiving a request, it dispatches a worker who is instructed to call in
when the job is completed. Historical data indicate that the time from when the server
is dispatched until he or she calls is a normal random variable with mean 62 minutes
and standard deviation 24 minutes. To keep aware of any changes in this distribution,